Gradient Flows as Optimal Controlled Evolutions: From Rn to Wasserstein product spaces

TL;DR

This paper unifies gradient flows and optimal control in Euclidean and Wasserstein spaces, providing new characterizations and insights, with potential applications in controlling microrobotic swarms.

Contribution

It introduces a novel variational framework linking gradient flows with optimal control in Wasserstein spaces, extending classical Euclidean results to more complex settings.

Findings

Gradient descent as optimal controlled evolution in Rn.

Wasserstein gradient flow characterized by a similar variational principle.

Application framework for controlling microrobotic swarms.

Abstract

We show that the continuous-time gradient descent in Rn can be viewed as an optimal controlled evolution for a suitable action functional; a similar result holds for stochastic gradient descent. We then provide an analogous characterization for the Wasserstein gradient flow of the (relative) entropy, with an action that mirrors the classical case where the Euclidean gradient is replaced by the Wasserstein gradient of the relative entropy. In the small-step limit, these continuous-time actions align with the Jordan Kinderlehrer Otto scheme. Next, we consider gradient flows for the relative entropy over a Wasserstein product space-a study motivated by the stochastic-control formulation of Schrodinger bridges. We characterize the product-space steepest descent as the solution to a variational problem with two control velocities and a product-space Wasserstein gradient, and we show that the induced fluxes in the two components are equal and opposite. This framework suggests applications to the optimal control evolution of microrobotic swarms that can communicate their present distribution to the other swarm.